Beyond nothingness in the formation and functional relevance of voids in polymer films

Voids—the nothingness—broadly exist within nanomaterials and impact properties ranging from catalysis to mechanical response. However, understanding nanovoids is challenging due to lack of imaging methods with the needed penetration depth and spatial resolution. Here, we integrate electron tomography, morphometry, graph theory and coarse-grained molecular dynamics simulation to study the formation of interconnected nanovoids in polymer films and their impacts on permeance and nanomechanical behaviour. Using polyamide membranes for molecular separation as a representative system, three-dimensional electron tomography at nanometre resolution reveals nanovoid formation from coalescence of oligomers, supported by coarse-grained molecular dynamics simulations. Void analysis provides otherwise inaccessible inputs for accurate fittings of methanol permeance for polyamide membranes. Three-dimensional structural graphs accounting for the tortuous nanovoids within, measure higher apparent moduli with polyamide membranes of higher graph rigidity. Our study elucidates the significance of nanovoids beyond the nothingness, impacting the synthesis‒morphology‒function relationships of complex nanomaterials.

Previous simulation efforts on PA membrane formation are mostly based on atomistic models.In a study conducted by Kolev et al. 1 , atomistic PA membranes were generated to provide insights into the mechanisms involved in the formation, hydration, and functioning of membranes during the interfacial polymerization (IP) processes.To simulate this, they initially set up a simulation box containing MPD and TMC monomers, along with a few TMC/MPD dimers acting as initial clusters.Following the initial setup, the monomers in the simulation were allowed to react exclusively with the growing clusters, while preventing reactions between monomers.With the formation of each new bond, a monomer of the same type as the one that reacted was introduced into the simulation box.This introduction occurred randomly within the vacant space not occupied by the van der Waals volume of the polymeric clusters.This simulation procedure closely resembles the stoichiometrically balanced diffusion of monomers from adjacent solutions into the reaction zone, as observed in IP processes, although the size of the simulation box is too small to capture nanomorphology.
Li et al. 2 performed equilibrium molecular dynamics (EMD) simulations to simulate the construction process of PA membranes at an atomic level.They investigated the morphological transformation and mass transport through the polymer network by varying the stoichiometry of the monomers.In their study, MPD and TMC molecules were initially placed in a cubic cell using a heuristic algorithm to ensure randomization.To achieve crosslinking between MPD and TMC, they utilised an update to the reaction radius.Specifically, if the nitrogen atom in a free amine group came within 3.25 Å of a carbonyl carbon atom in a free acyl chloride group, a molecular topology transformation was triggered.This transformation involved the formation of an amide bond while simultaneously removing any excess hydrogen and chlorine atoms present.The crosslinking process involved continuous updates to the atomic radii of the reaction until reaching either the maximum C-N cutoff distance or the desired conversion target.Different MPD:TMC ratios were considered, corresponding to different stoichiometric ratios based on the amine:acyl chloride functional group molar ratio.He et al. 3 performed non-equilibrium molecular dynamics (NEMD) simulations to investigate the effect of different manufacturing methods on the performance of PA membranes.For IP in particular, a fixed 3:2 MPD:TMC ratio was used for the simulations.The reacted atoms were identified and arranged into TMC and MPD monomers.Each monomer type provided specific potential reaction sites.These monomers were then filled into separate 3D-periodic cells of equal cross-section size.The two cells were assembled along the direction normal to the cross-section, creating a composite with an interfacial layer between the MPD and TMC layers.The cross-linking reaction occurred randomly and was restrained to the MPD/TMC interface, simulating the IP process.
In a study by Shen et al. 4 , NEMD simulations were used to study the atomic-scale transport of water, ions, and small organic solutes in a commonly used membrane.In this process, the TMC and MPD monomers were randomly moved in a computational box.When the functional groups of the monomers responsible for cross-linking were within a specified distance from each other, an amide bond was formed, subsequently building the polymeric structure.The aim was to mimic the variability observed in actual polymerization processes, considering the inherent randomness involved.

Supplementary Note 3: CGMD simulations of varying monomer diffusion rates
We incorporated three additional simulations to investigate the impact of MPD diffusion rates on stoichiometry within the reaction zone and the final PA membrane density.To optimise computational efficiency, we utilised the model and simulation setup previously established by Muscatello et al. 5 .The simulation box dimensions are (10 × 10 × 200) nm 3 , with a TMC:MPD concentration ratio of 1:1.Subsequently, to characterise three distinct monomer diffusion rates, we arbitrarily varied the mass input for the MPD monomer as 0.3, 1.0, and 3.0 times the original molecular mass 6 .Note that the diffusion rate of MPD monomers in hexane 7 is 10 -6 cm 2 s -1 , while in our CGMD simulations, the diffusion rate of CG-MPD monomers is 1.265×10 -6 cm 2 s -1 , a result of using implicit solvent conditions.We observed that diverse diffusion rates affect the local membrane configuration during the reaction process, resulting in denser membrane structures when the rate of diffusion of MPD monomer is low and less dense membrane structures with a higher rate of diffusion compared to the original monomer diffusion rate.In other words, the different diffusion rates of the amine monomers affectthe monomer stoichiometry in the reaction zone as observed with the distinct local membrane configuration (Supplementary Figs.11,12).

Supplementary Note 4: Derivation of solvent permeance fittings
We used the Spiegler-Kedem model as the transport equation for the permeance fitting as shown in equation (1) below, where Jv is the volumetric flux, Lp is the hydraulic permeance, ΔP is the transmembrane pressure, σ is the reflection coefficient and Δπ is the osmotic pressure.
In the Spiegler-Kedem model, Lp is given by the solvent-membrane permeability (Pm) and the thickness of the membrane (δ), which is used as the basis to derive our fitting (equation ( 2)).
To determine the nominal membrane thickness, the membrane was categorized into three distinct regions, which are depicted in Supplementary Fig. 13.Region 1 is the thin open void, region 2 is the featureless base layer, and region 3 is the thick closed void section.
Tomographic reconstructions were used to calculate the top (AT) and bottom (AB) surface areas of the PA membrane, the surface area of open voids (AOV) and the surface area of closed voids (ACV).
The geometric area (AG) is the projection area of the tomographic reconstruction (Supplementary Table 5).
Two approaches are used to fit the experimental methanol permeance.The first fitting (Approach 1) used open and closed void areas, and the local membrane thicknesses measured using tomographic reconstruction.
For Approach 1, the surface areas of region 1 and region 3 are given by AOV and ACV respectively, and the surface area of the featureless region 2 (AF) is given by equation (3).
Using the surface area data from tomography, the area percentage for each region as shown in equations ( 4), ( 5) and ( 6) where region 1, region 2 and region 3 are represented by %Aopen, %Aflat and %Aclosed, respectively.
%  =     +   +   (4) %  =     +   +   (6)   Three assumptions are driven by observations from the 3D tomographic reconstructions of PA membranes, thickness mapping and void reconstructions; (i) open void regions are surrounded by t1 and t2 thickness, (ii) closed void regions are enclosed by one layer of t2 thickness and another layer of t3 thickness at the base, and (iii) the flat membrane region is of t2 thicknesses.
To estimate the nominal closed void thickness (region 3),   , the thickness maxima enclosing the top void region, t3, was added to the thickness maxima representing the bottom surface thickness t2 (equation ( 8)).
To calculate the nominal thickness (δnom,1) from these three distinct regions, a harmonic mean of each region's thickness weighted by the percentage of area of that region was used as shown in equation (10).
The second fitting (Approach 2) used membrane thicknesses derived from AFM measurements (δnom,2) as shown in Supplementary Fig. 14.
For both approaches, the permeability (Pm) was then calculated given the nominal membrane thickness and experimentally determined solvent permeance (Lexp,k) with the following equation: The material properties of the membranes were assumed to be dependent on the mass.A permeability constant (μ) was defined to isolate the effect of polymer density (equation ( 12)), as the solvent-membrane permeability is inversely proportional to membrane density.Note that the density is a linear function of the degree of crosslinking within our range of interest (41% < DOC < 98%) with an R 2 of 0.98, thus the density was calculated using DOC based on previous work 8,9 .
The permeability constant for each PA membrane was averaged to form an average permeability constant (μavg), and used to calculate the solvent permeance (Lfit,k) as follows (equation ( 13)).

Table 7 . Additional GT parameters calculated for PA membrane skeletons.
12V and AVS are used to quantify the complexity of a graph.AVV assigns a numerical value to each node of the graph by considering the degree of all nodes connected to the node of interest, weighted against the distance. 11VS is the sum of AVV values of all the nodes of the graph with inequivalent symmetry.12 n : number of nodes e : the number of branches k : degree of a given node l : distance between two nodes